Question: Rewrite the function by completing the square. $g(x)= 2 x^{2} -7 x +5$ $g(x)=$
Solution: $\begin{aligned} g(x)&=2 x^2 -7 x +5 \\\\ &=2 \left(x^2 -\dfrac{7}{2} x\right) +5 \end{aligned}$ Now we want to complete $x^2 -\dfrac{7}{2} x$ into a perfect square. To do that, we should add $\left(\dfrac{{-\frac{7}{2}}}{2}\right)^2={\dfrac{49}{16}}$ to it: $x^2{-\dfrac{7}{2}}x+{\dfrac{49}{16}}=\left(x -\dfrac{7}{4}\right)^2$ We add ${\dfrac{49}{16}}$ inside the parentheses, and subtract ${2}\cdot{\dfrac{49}{16}}$ outside them, to keep the expression equivalent. $\begin{aligned} &\phantom{=}{2} \left(x^2 -\dfrac{7}{2} x\right) +5 \\\\ &={2}\left(x^2 -\dfrac{7}{2} x+{\dfrac{49}{16}}\right) +5 -{2}\cdot{\dfrac{49}{16}} \\\\ &=2 \left(x -\dfrac{7}{4}\right)^2 +5 -\dfrac{49}{8} \\\\ &=2 \left(x -\dfrac{7}{4}\right)^2 -\dfrac{9}{8} \end{aligned}$ In conclusion, the function after completing the square is written as: $g(x)=2 \left(x -\dfrac{7}{4}\right)^2 -\dfrac{9}{8}$